A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

26 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation
and
∂ ∂u
------- ⎛ 1
ν ------- ⎞ ∂ ∂u
------- ⎛ 1
∂x
⎝ t ν ------- ⎞ ∂ ∂u1
+
1 ∂x
⎠
1 ∂x
⎝ t + ------- ⎛ν------- ⎞
2 ∂x
⎠
2 ∂x
⎝ t
3 ∂x
⎠
3
(2.23)
∂ ∂u
------- ⎛ 2
ν ------- ⎞ ∂ ∂u
------- ⎛ 2
∂x
⎝ t
ν ------- ⎞ ∂ ∂u2
+
1 ∂x
⎠
1 ∂x
⎝ t + ------- ⎛ν------- ⎞
2 ∂x
⎠
2 ∂x
⎝ t . (2.24)
3 ∂x
⎠
3
The form of 2.23 and 2.24 does not preserve the symmetry of the turbulent stress tensor because the right sides of
– u1 ′ u′ 2 = νt( ∂u1 ⁄ ∂x2)
and – u2 ′ u′ 1 = νt( ∂u2 ⁄ ∂x1)
are no longer equal as they should be. For this reason, the form of
2.20 and 2.21 sometimes is preferred. It turns out, however, that the loss of symmetry in the stress tensor generally is not a serious
concern for estuarine modeling.
When 2.20 and 2.21 or 2.23 and 2.24 is introduced into equation 2.15, the problem of turbulence closure is shifted to that of
defining the distribution of the eddy viscosity. Unlike the molecular viscosity, the eddy viscosity is not a fluid property but instead
varies throughout the flow field as a function of the length and velocity scales of the turbulent eddies. Because the largest-scale
eddies are primarily responsible for the turbulent transport of momentum and salt, these largest eddies influence most the magni-
tude of v t. The geometry of the estuary determines to a great extent the size of the largest eddies. Because of the striking disparity
between the length scales of the horizontal and vertical dimensions of most estuaries, the intensity and size of the horizontal eddies
are much greater than those of the vertical eddies. The turbulent motions under these conditions are said to be anisotropic (direction
dependent). It is typical to define separate horizontal and vertical eddy viscosities, A H and A V , to account for the anisotropy. The
turbulent stress terms in the form of 2.23 and 2.24 then become
and
∂ ∂u
------- ⎛ 1
A ------- ⎞ ∂ ∂u
------- ⎛ 1
∂x
⎝ H
A ------- ⎞ ∂ ∂u1
+
1 ∂x
⎠
1 ∂x
⎝ H + ------- ⎛A------- ⎞
2 ∂x
⎠
2 ∂x
⎝ V
(2.25)
3 ∂x
⎠
3
∂ ∂u
------- ⎛ 2
A -------- ⎞ ∂ ∂u
------- ⎛ 2
∂x ⎝ H
A -------- ⎞ ∂ ∂u2 +
1 ∂x ⎠
1 ∂x ⎝ H + ------- ⎛A-------- ⎞
2 ∂x ⎠
2 ∂x ⎝ V . (2.26)
3 ∂x ⎠
3
For estuaries, a possible range of magnitudes is 10 3 to 10 6 cm 2 /s for A H and 1 to 500 cm 2 /s for A V.
Because of the large horizontal extent of flow in most estuaries, the horizontal variations in mean-flow quantities generally
are much more gradual than the vertical variations. As a result, the four terms in 2.25 and 2.26 involving horizontal gradients of
mean velocities ordinarily are much smaller than the two terms involving vertical gradients. (This will be demonstrated also in
Section 2.5 using scale analysis.) Each of the terms in 2.25 and 2.26 represent momentum diffusion due to turbulence in addition
to their interpretation as stresses. Horizontal momentum diffusion generally is less significant in estuaries than vertical momentum
diffusion.